It is suitable to compare our Hamiltonian Dysthe equation (REF )
with another related model that has recently been derived by [1]}, [2]} in the same physical setting.
Note that these authors additionally considered surface tension but we will only examine the gravity-wave version of their model.
Moreover, because they expressed their model in a form that contains a first derivative in time as well as a mixed derivative
in space and time (see Eq. 2.37 in [1]}), we find it more appropriate to rewrite it in a more standard form
with a single time derivative (as it is typically so for the Dysthe equation [4]}) to allow for a fairer comparison.
We also take into account the fact that vorticity in the mathematical formulation used by [1]}
is defined as the opposite of ours.
The resulting model for the first-harmonic envelope \(\eta _1\)  is given by
\( \partial _t \eta _1 & = & -c_g \partial _x \eta _1 - i \frac{c_g^2}{d_0} \partial _x^2 \eta _1+ 2 \frac{c_g^3}{d_0^2} \partial _x^3 \eta _1- i \frac{\widetilde{\alpha }_0}{d_0} |\eta _1|^2 \eta _1 \nonumber \\& & - \frac{\widetilde{\alpha }_1}{d_0} |\eta _1|^2 \partial _x \eta _1- \frac{\widetilde{\alpha }_2}{d_0} \eta _1^2 \partial _x \overline{\eta }_1+ i \frac{\widetilde{\alpha }_3}{d_0} \eta _1 {\rm H}\partial _x |\eta _1|^2 \,,\) 
